1.1: Starting with Sentences

We begin the study of logic by building a precise logical language. This will allow us to do at least two things: first, to say some things more precisely than we otherwise would be able to do; second, to study reasoning. We will use a natural language—English—as our guide, but our logical language will be far simpler, far weaker, but more rigorous than English.

We must decide where to start. We could pick just about any part of English to try to emulate: names, adjectives, prepositions, general nouns, and so on. But it is traditional, and as we will see, quite handy, to begin with whole sentences. For this reason, the first language we will develop is called “the propositional logic”.It is also sometimes called “the sentential logic” or even “the sentential calculus”. These all mean the same thing: the logic of sentences. In this propositional logic, the smallest independent parts of the language are sentences (throughout this book, I will assume that sentences and propositions are the same thing in our logic, and I will use the terms “sentence” and “proposition” interchangeably).

There are of course many kinds of sentences. To take examples from our natural language, these include:

What time is it?

Open the window.

Damn you!

I promise to pay you back.

It rained in Central Park on June 26, 2015.

We could multiply such examples. Sentences in English can be used to ask questions, give commands, curse or insult, form contracts, and express emotions. But, the last example above is of special interest because it aims to describe the world. Such sentences, which are sometimes called “declarative sentences”, will be our model sentences for our logical language. We know a declarative sentence when we encounter it because it can be either true or false.